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1.
Finite Projective Geometries and Classification of the Weight Hierarchies of Codes (Ⅰ) 总被引:1,自引:0,他引:1
WenDeCHEN TorleivKLφVE 《数学学报(英文版)》2004,20(2):333-348
The weight hierarchy of a binary linear [n,κ] code C is the sequence (d1,d2,...,dκ), where dr is the smallest support of an r-dimensional subcode of C. The codes of dimension 4 are collected in classes and the possible weight hierarchies in each class is determined by finite projective geometries.The possible weight hierarchies in class A, B, C, D are determined in Part (Ⅰ). The possible weight hierarchies in class E, F, G, H, I are determined in Part (Ⅱ). 相似文献
2.
Mattias Svanström 《Designs, Codes and Cryptography》1999,18(1-3):223-229
We construct a class of perfect ternary constant-weight codes of length 2
r
, weight 2
r
-1 and minimum distance 3. The codes have
codewords. The construction is based on combining cosets of binary Hamming codes. As a special case, for r=2 the construction gives the subcode of the tetracode consisting of its nonzero codewords. By shortening the perfect codes, we get further optimal codes. 相似文献
3.
The weight hierarchy of a linear [n,k;q] code C over GF(q) is the sequence (d1,d2,...,dk) where dr is the smallest support of an r–dimensional subcode of C. By explicit construction, it is shown that if a sequence (a1,a2,...,ak) satisfies certain conditions, then it is the weight hierarchy of a code satisfying the chain condition. 相似文献
4.
We show that almost all codes satisfy an antichain condition. This states that the minimum length of a two dimensional subcode of a code C increases if the subcode is constrained to contain a minimum weight codeword. In particular, almost no code satisfies the chain condition. In passing, we study the typical behaviour of codes with respect to generalized distances and show that almost all lie on a generalized Varshamov-Gilbert bound. 相似文献
5.
《Finite Fields and Their Applications》2001,7(2):355-371
We build a class of codes using hermitian forms and the functional trace code. Then we give a general expression of the rth minimum distance of our code and compute general bounds for the weight hierarchy by using exponential sums. We also get the minimum distance and calculate the rth generalized Hamming weight dr in some special cases. 相似文献
6.
《Discrete Mathematics》2001,221(1-3):171-187
The difference g2−d2 for a q-ary linear [n,3,d] code C is studied. Here d2 is the second generalized Hamming weight, that is, the smallest size of the support of a 2-dimensional subcode of C; and g2 is the second greedy weight, that is, the smallest size of the support of a 2-dimensional subcode of C which contains a codeword of weight d. For codes of dimension 3, it is shown that the problem is essentially equivalent to finding certain weighting of the points in the projective plane, and weighting which give the maximal value of g2−d2 are determined in almost all cases. In particular max(g2−d2) is determined in all cases for q⩽9. 相似文献
7.
We use methods of Mortimer [19] to examine the subcodes spanned by minimum-weight vectors of the projective generalized Reed-Muller codes and their duals. These methods provide a proof, alternative to a dimension argument, that neither the projective generalized Reed-Muller code of order r and of length
over the finite field F
q
of prime-power order q, nor its dual, is spanned by its minimum-weight vectors for 0<r<m–1 unless q is prime. The methods of proof are the projective analogue of those developed in [17], and show that the codes spanned by the minimum-weight vectors are spanned over F
q
by monomial functions in the m variables. We examine the same question for the subfield subcodes and their duals, and make a conjecture for the generators of the dual of the binary subfield subcode when the order r of the code is 1. 相似文献
8.
Dmitrii Yu. Nogin 《Finite Fields and Their Applications》1999,5(4):409
We show that the minimum r-weight dr of an anticode can be expressed in terms of the maximum r-weight of the corresponding code. As examples, we consider anticodes from homogeneous hypersurfaces (quadrics and Hermitian varieties). In a number of cases, all differences (except for one) of the weight hierarchy of such an anticode meet an analog of the generalized Griesmer bound. 相似文献
9.
Todd D. Vance 《Designs, Codes and Cryptography》2000,19(1):27-43
The weight distribution of GRM (generalized Reed-Muller) codes is unknown in general. This article describes and applies some new techniques to the codes over F3. Specifically, we decompose GRM codewords into words from smaller codes and use this decomposition, along with a projective geometry technique, to relate weights occurring in one code with weights occurring in simpler codes. In doing so, we discover a new gap in the weight distribution of many codes. In particular, we show there is no word of weight 3m–2 in GRM3(4,m) for m>6, and for even-order codes over the ternary field, we show that under certain conditions, there is no word of weight d+, where d is the minimum distance and is the largest integer dividing all weights occurring in the code. 相似文献
10.
Tuvi Etzion 《组合设计杂志》2008,16(2):137-151
A doubly constant weight code is a binary code of length n1 + n2, with constant weight w1 + w2, such that the weight of a codeword in the first n1 coordinates is w1. Such codes have applications in obtaining bounds on the sizes of constant weight codes with given minimum distance. Lower and upper bounds on the sizes of such codes are derived. In particular, we show tight connections between optimal codes and some known designs such as Howell designs, Kirkman squares, orthogonal arrays, Steiner systems, and large sets of Steiner systems. These optimal codes are natural generalization of Steiner systems and they are also called doubly Steiner systems. © 2007 Wiley Periodicals, Inc. J Combin Designs 16: 137–151, 2008 相似文献
11.
Hao Chen 《中国科学A辑(英文版)》2009,52(10):2171-2176
Property testing was initially studied from various motivations in 1990’s. A code C GF (r)n is locally testable if there is a randomized algorithm which can distinguish with high possibility the codewords from a vector essentially far from the code by only accessing a very small (typically constant) number of the vector’s coordinates. The problem of testing codes was firstly studied by Blum, Luby and Rubinfeld and closely related to probabilistically checkable proofs (PCPs). How to characterize locally te... 相似文献
12.
The notion of a shadow of a self-dual binary code is generalized to self-dual codes over
4. A Gleason formula for the symmetrized weight enumerator of the shadow of a Type I code is derived. Congruence properties of the weights follow; this yields constructions of self-dual codes of larger lengths. Weight enumerators and the highest minimum Lee, Hamming, and Euclidean weights of Type I codes of length up to 24 are studied. 相似文献
13.
Harold N Ward 《Journal of Combinatorial Theory, Series A》1976,21(2):253-255
Gleason's theorem gives the general form of the weight enumerator of a linear binary self-dual code; it is a linear combination with integral coefficients of certain polynomials. When the subcode of words whose weights are multiples of 4 is not the whole code, the MacWilliams identities applied to that subcode yield divisibility conditions on those coefficients. The conditions show that there are no further extremal codes, for Case 1 in the sense of Mallows and Sloane, than the ones known. 相似文献
14.
Truong et al. [7]proved that the weight distribution of a binary quadratic residue code C with length congruent to −1 modulo 8 can be determined by the weight distribution of a certain subcode of C containing only one-eighth of the codewords of C. In this paper, we prove that the same conclusion holds for any binary quadratic residue codes. 相似文献
15.
Ederlina G. Nocon 《Designs, Codes and Cryptography》2003,30(3):301-323
We consider here the construction of Type II codes over the abelian group Z4×Z4. The definition of Type II codes here is based on the definitions introduced by Bannai [2]. The emphasis is given on the construction of these types of codes over the abelian group Z4×Z4 and in particular, the methods applied by Gaborit [7] in the construction of codes over Z4 was extended to four different dualities with their corresponding weight functions (maps assigning weights to the alphabets of the code). In order to do this, we use the flattened form of the codes and construct binary codes analogous to the ones applied to Z4 codes. Since each duality generates more than one weight function, we focus on those weights satisfying the squareness property. Here, by the squareness property, we mean that the weight function wt assigns the weight 0 to the Z4×Z4 elements (0, 0),(2, 2) and the weight 4 to the elements (0, 2) and (2, 0). The main results of this paper are focused on the characterization of these codes and provide a method of construction that can be applied in the generation of such codes whose weight functions satisfy the squareness property. 相似文献
16.
In this paper, codes over F5 with parameters [36, 18, 12], [48, 24, 15], [60, 30, 18], [64, 32, 18] and [76, 38, 21] which improve the previously known bounds on the minimum weight for linear codes over F5 are constructed from conference matrices. Through shortening and truncating, the above codes give numerous new codes over F5 which improve the previously known bounds on minimum weights. 相似文献
17.
A code c is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from c. The minimum size of such a c is denoted by c
r(X). Answering a question of Hämäläinen et al. [10], we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n
0(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}n. We prove that c
r(B
n(0, r + 2)) = 1 i r + 1 ( (n + i – 1) / (r + 1) 2) + n / (r + 1) and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts. 相似文献
18.
Veerle Fack Szabolcs L. Fancsali L. Storme Geetrui Van de Voorde Joost Winne 《Designs, Codes and Cryptography》2008,46(1):25-43
We study codewords of small weight in the codes arising from Desarguesian projective planes. We first of all improve the results
of K. Chouinard on codewords of small weight in the codes arising from PG(2, p), p prime. Chouinard characterized all the codewords up to weight 2p in these codes. Using a particular basis for this code, described by Moorhouse, we characterize all the codewords of weight
up to 2p + (p−1)/2 if p ≥ 11. We then study the codes arising from . In particular, for q
0 = p prime, p ≥ 7, we prove that the codes have no codewords with weight in the interval [q + 2, 2q − 1]. Finally, for the codes of PG(2, q), q = p
h
, p prime, h ≥ 4, we present a discrete spectrum for the weights of codewords with weights in the interval [q + 2, 2q − 1]. In particular, we exclude all weights in the interval [3q/2, 2q − 1].
Geertrui Van de Voorde research is supported by the Institute for the Promotion of Innovation through Science and Technology
in Flanders (IWT-Vlaanderen)
Joost Winne was supported by the Fund for Scientific Research - Flanders (Belgium). 相似文献
19.
Certain
-modules related to the kernels ofincidence maps between types in the poset defined by the natural productorder on the set of n-tuples with entries from {1,
,m} are studied as linear codes (whencoefficients are extended to an arbitrary field K). Theirdimensions and minimal weights are computed. The Specht modules areextremal among these submodules. The minimum weight codewords of theSpecht module are shown to be scalar multiples of polytabloids. Ageneralization of t-design arising from the natural permutationS
n-modules labelled by partitions with mparts is introduced. A connection with Reed-Muller codes is noted and acharacteristic free formulation is presented. 相似文献
20.
Nejib Zaguia 《Order》1987,4(3):257-267
A bump (x
i,x
i+1) occurs in a linear extension L={x
1<...n} of a poset P, if x
ii+1 in P. L. is greedy if x
ij for every j>i, whenever (x
i
x
i+1) in a bump in L. The purpose of this paper is to give a characterization of all greedy posets. These are the posets for which every greedy linear extension has a minimum number of bumps.This research (Math/1406/31) was supported by the Research Center, College of Science, King Saud University, Riyadh, Saudi Arabia. 相似文献